Exploring the limits of multifunctionality in tunable networks

Nature is rife with networks that are functionally optimized to propagate localized inputs to perform specific tasks. For example, allosteric proteins globally change conformation upon the binding of a ligand, controlling the activity of a distant active site. As another example, the vascular network in the brain can reroute blood flow to enhance oxygen levels to locally support active neurons. Whether via genetic evolution or dynamic adaptation, many networks create functionality by locally tuning edge properties. To explore this behavior, we optimize both mechanical and flow networks to perform specific functions by adding and removing edges. We define a single function as a tuned response (strain or pressure drop, respectively) of a single target edge when another specified part of the network is activated (similarly via strain or pressure drop, respectively). Using structures generated via such optimization, we answer the question of how many simultaneous functions a given network can be programmed to fulfill. We find that both types of networks display similar phase transitions in the number of targets that can be tuned, implying that both tuning problems can be understood in the context of a broader class of constraint-satisfaction problems.